L(s) = 1 | − 3·5-s − 7-s + 3·9-s + 11-s + 6·17-s + 19-s + 3·23-s + 5·25-s + 12·29-s − 2·31-s + 3·35-s + 12·41-s + 18·43-s − 9·45-s + 3·47-s − 6·49-s + 2·53-s − 3·55-s + 12·59-s − 11·61-s − 3·63-s + 8·67-s + 12·71-s + 15·73-s − 77-s + 12·79-s + 14·83-s + ⋯ |
L(s) = 1 | − 1.34·5-s − 0.377·7-s + 9-s + 0.301·11-s + 1.45·17-s + 0.229·19-s + 0.625·23-s + 25-s + 2.22·29-s − 0.359·31-s + 0.507·35-s + 1.87·41-s + 2.74·43-s − 1.34·45-s + 0.437·47-s − 6/7·49-s + 0.274·53-s − 0.404·55-s + 1.56·59-s − 1.40·61-s − 0.377·63-s + 0.977·67-s + 1.42·71-s + 1.75·73-s − 0.113·77-s + 1.35·79-s + 1.53·83-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1132096 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1132096 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(2.178124223\) |
\(L(\frac12)\) |
\(\approx\) |
\(2.178124223\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $\Gal(F_p)$ | $F_p(T)$ |
---|
bad | 2 | | \( 1 \) |
| 7 | $C_2$ | \( 1 + T + p T^{2} \) |
| 19 | $C_2$ | \( 1 - T + T^{2} \) |
good | 3 | $C_2$ | \( ( 1 - p T + p T^{2} )( 1 + p T + p T^{2} ) \) |
| 5 | $C_2^2$ | \( 1 + 3 T + 4 T^{2} + 3 p T^{3} + p^{2} T^{4} \) |
| 11 | $C_2^2$ | \( 1 - T - 10 T^{2} - p T^{3} + p^{2} T^{4} \) |
| 13 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 17 | $C_2^2$ | \( 1 - 6 T + 19 T^{2} - 6 p T^{3} + p^{2} T^{4} \) |
| 23 | $C_2^2$ | \( 1 - 3 T - 14 T^{2} - 3 p T^{3} + p^{2} T^{4} \) |
| 29 | $C_2$ | \( ( 1 - 6 T + p T^{2} )^{2} \) |
| 31 | $C_2^2$ | \( 1 + 2 T - 27 T^{2} + 2 p T^{3} + p^{2} T^{4} \) |
| 37 | $C_2^2$ | \( 1 - p T^{2} + p^{2} T^{4} \) |
| 41 | $C_2$ | \( ( 1 - 6 T + p T^{2} )^{2} \) |
| 43 | $C_2$ | \( ( 1 - 9 T + p T^{2} )^{2} \) |
| 47 | $C_2^2$ | \( 1 - 3 T - 38 T^{2} - 3 p T^{3} + p^{2} T^{4} \) |
| 53 | $C_2^2$ | \( 1 - 2 T - 49 T^{2} - 2 p T^{3} + p^{2} T^{4} \) |
| 59 | $C_2^2$ | \( 1 - 12 T + 85 T^{2} - 12 p T^{3} + p^{2} T^{4} \) |
| 61 | $C_2^2$ | \( 1 + 11 T + 60 T^{2} + 11 p T^{3} + p^{2} T^{4} \) |
| 67 | $C_2^2$ | \( 1 - 8 T - 3 T^{2} - 8 p T^{3} + p^{2} T^{4} \) |
| 71 | $C_2$ | \( ( 1 - 6 T + p T^{2} )^{2} \) |
| 73 | $C_2^2$ | \( 1 - 15 T + 152 T^{2} - 15 p T^{3} + p^{2} T^{4} \) |
| 79 | $C_2^2$ | \( 1 - 12 T + 65 T^{2} - 12 p T^{3} + p^{2} T^{4} \) |
| 83 | $C_2$ | \( ( 1 - 7 T + p T^{2} )^{2} \) |
| 89 | $C_2^2$ | \( 1 - 8 T - 25 T^{2} - 8 p T^{3} + p^{2} T^{4} \) |
| 97 | $C_2$ | \( ( 1 + 6 T + p T^{2} )^{2} \) |
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\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{4} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−10.22769511629674240766279792076, −9.546790746165948065312112632774, −9.205301490955663687708904899765, −9.143902950389557475859413082157, −8.103981915077983012699863857576, −8.079547037625268987826141655934, −7.69120072171708816721776709600, −7.26290838794624075797398516820, −6.77948139946002996424791044319, −6.45706981920482044028320982333, −5.92142470090431956204746010342, −5.20592449328388013626858252258, −4.92954314549396733363910974375, −4.27130214828303630181140653988, −3.80196400597995206373404637535, −3.66477463995464936433010421017, −2.79342580269841575725242900349, −2.42428177243399102631439844589, −0.998838402120492877450497847243, −0.945637548330351867245348268594,
0.945637548330351867245348268594, 0.998838402120492877450497847243, 2.42428177243399102631439844589, 2.79342580269841575725242900349, 3.66477463995464936433010421017, 3.80196400597995206373404637535, 4.27130214828303630181140653988, 4.92954314549396733363910974375, 5.20592449328388013626858252258, 5.92142470090431956204746010342, 6.45706981920482044028320982333, 6.77948139946002996424791044319, 7.26290838794624075797398516820, 7.69120072171708816721776709600, 8.079547037625268987826141655934, 8.103981915077983012699863857576, 9.143902950389557475859413082157, 9.205301490955663687708904899765, 9.546790746165948065312112632774, 10.22769511629674240766279792076